TOWARD THE FORMAL THEORY OF (, n)-categories

Transcription

1 TOWRD THE FORML THEORY OF (, n)-ctegories EMILY RIEHL (JOINT WORK WITH DOMINIC VERITY) bstract. Formal category theory reers to a commonly applicable ramework (i) or deining standard categorical strctres monads, adjnctions, limits, the Yoneda embedding, Kan extensions and (ii) in which the classical proos can be sed to establish the expected relationships between these notions: e.g. that right adjoints preserve limits. One sch ramework is a 2- category eqipped with a bicategory o modles. ( modle or pronctor rom a category to a category is a nctor op Set, or instance hom: op Set.) In [RV1], we show the basic category theory o qasi-categories can be developed ormally in a strict 2-category, the homotopy 2-category o qasicategories. main point is that certain weak 2-limits present in this 2-category, particlarly comma objects, ene niversal properties p to the appropriate notion o eqivalence or qasi-categories. n important eatre o these ormal deinitions and proos is that they apply representably in other higher homotopical contexts, inclding Rezk objects (e.g., complete Segal spaces). In the qasi-categorical context, we are reprising the ondational work pioneered by Joyal, Lrie, and others. Or aim is to develop new tools to prove rther theorems, bt an important side beneit is that this work applies eqally to other models. The aorementioned comma objects are precisely those modles that are represented by ordinary nctors. In work in progress, we have developed a general theory o modles between qasi-categories, which is robst enogh to spport a complete ormal category theory. (Modles appear nder the gise o correspondences in Lrie s work, bt or presentation, as two-sided discrete ibrations, is dierent.) This allows s to prove, or instance, the amiliar (co)limit ormla or pointwise Kan extensions. t present, these new reslts do not immediately translate to other lavors o (, 1)-categories, or to (, n)-categories, becase there is one key technical property (a homotopy exponentiability criterion or maps) that we prove in speciic reerence to the qasi-categorical model. In what ollows, we explain some o the basic ideas behind ormal category theory (how comma objects are sed to ene categorical strctres), describe the homotopy expontentiability criterion, and explore tre vistas. Overview Speaking loosely, an (, n)-category is a weak higher category with objects; 1-morphisms, 2-morphisms, 3-morphisms, and so on in each positive dimension; and with the property that all morphisms above dimension n are weakly invertible. For instance, the points in a space can be regarded as the objects o an (, 0)- category whose 1-morphisms are paths, 2-morphisms are homotopies, 3-morphisms are homotopies between homotopies, and so on. Thanks to a lot o hard work Date: September 15, 2014 Topologie Workshop Oberwolach. 1

2 2 EMILY RIEHL (JOINT WORK WITH DOMINIC VERITY) with contribtions made by many people, there are a plethora o models o (, n)- categories, which, by even more hard work, have been shown to be eqivalent, in a sitable sense. Or aim is to provide the tools necessary to do something with (, n)-categories: to nderstand what it means or some (, n)-category to have limits o a particlar shape and what it means or a nctor between (, n)-categories to have a right adjoint. These notions are not nrelated: as a test that the right deinitions have been identiied, one shold be able to prove that right adjoints preserve limits. This sort o work has been done or one particlar model o (, 1)-categories, namely the qasi-categories stdied by Joyal, which are called -categories by Lrie. However, the deinitions and proos are highly technical and cannot easily be generalized to other models o (, 1)-categories or to (, n)-categories. To achieve this, we se ideas rom ormal category theory to gide or deinitions and simpliy or proos. In so doing, we recover the category theory o qasi-categories and can immediately generalize most o or reslts to other higher category contexts. asic ormal category theory The simplest ramework or ormal category theory is a strict 2-category that we will denote by K 2. The prototypical example might be the 2-category o categories, nctors, and natral transormations. Soon K 2 will be a 2-category whose objects are (, n)-categories, whose morphisms are nctors o sch, and whose 2-cells are homotopy classes o 1-simplices in appropriate hom-spaces, which we call natral transormations. djnctions and eqivalences. We will se single arrows to denote 1-cells and doble arrows to denote 2-cells. Deinition. n adjnction consists o objects, ; 1-cells :, : ; and 2-cells η : id, ɛ: id satisying the ollowing pair o identities η = ɛ ɛ 7 η = In a strict 2-category, any pasting diagram o 2-cells, sch as displayed in the preceding deinition, has a niqe composite 2-cell. The displayed pasting diagram is the 2-categorical ening o the so-called triangle identities, which assert that both pasting composites o the nit and conit 2-cells are identities. The standard proos o the ollowing reslts can be interpreted in a generic 2- category. Proposition. (i) ny two let adjoints to a common 1-cell are isomorphic. (ii) djnctions compose. In any 2-category, there is a standard notion o eqivalence between objects. Deinition. n eqivalence between a pair o objects and consists o a pair o 1-cells :, : and a pair o 2-cell isomorphisms id = and id =.

3 TOWRD THE FORML THEORY OF (, n)-ctegories 3 gain, the standard proo can be interpreted in a generic 2-category. Proposition. ny eqivalence can be promoted to an adjoint eqivalence. Limits and colimits. Now sppose that K 2 has some notion o exponentiation, an operation that deines 2-nctors ( ) : K 2 K 2 indexed by objects in some other category (sch as Cat or sset). I is an object o K 2, the object is thoght o as the object o -shaped diagrams in. s is standard, a morphism Y shold indce a map Y. We ll assme also that exponentiation by the terminal object is the identity; this gives rise to a constant diagram map const: or any diagram shape. In the examples we will consider, the provenance o sch a strctre is obvios, so I won t spell ot precisely what is necessary. Deinition. n object K 2 has -shaped limits i the 1-cell const: has a right adjoint lim:. Proposition. I and have -shaped limits, any right adjoint : preserves them. Proo. Sppose is let adjoint to. y binctoriality o exponentiation, the 1-cells const and const agree. Ths the right adjoints lim and lim are isomorphic. Remark. The lly general version o the previos proposition in which is assmed to have certain, bt not necessarily all, -shaped limits, and nothing is assmed a priori abot is no more diiclt to prove. The reason we have not presented it here is it reqires s to discss the general deinition o what it means or to have colimits o particlar -shaped diagrams, a property which is ened by an absolte right liting diagram in K 2. bsolte right liting diagrams are very easy to work with bt are most likely namiliar. Weak comma objects. Frther reslts are possible i the 2-category K 2 has weak comma objects. Deinition. Given g C, a weak comma object consists o the data g C g satisying a weak niversal property with three components: (1-cell indction): Given any 2-cell as displayed on the let c b C g = C g g

4 4 EMILY RIEHL (JOINT WORK WITH DOMINIC VERITY) there exists a 1-cell g so that the given 2-cell actors throgh the comma cone along this map. (2-cell indction): Deined similarly. (2-cell conservativity): ny 2-cell indced by a pair o 2-cell isomorphisms is itsel an isomorphism. s a corollary o these niversal properties, the 1-cells g indced by a given 2-cell orm a connected gropoid over the identities on c and b. Example. The ollowing special cases o weak comma objects are particlarly important: α β generalized element o, meaning a morphism, corresponds to a generalized element a: o, a generalized element b: o, together with a 2-cell b a. Proposition. I, then and are eqivalent over. Given a morphism b a, there is a standard ormla or the adjnct morphism o b a in terms o the nit and conit o the adjnction. This argment is precisely ened by the ollowing 2-categorical proo. Proo. y 1-cell indction, there are 1-cells w : and w : deined to satisy the identities: β ɛ w α w = = η α β y deinition, these 1-cells lie over. y the series o pasting identities w w = w = η β ɛ ɛ α α = α

5 TOWRD THE FORML THEORY OF (, n)-ctegories 5 ww and id are indced by the same 2-cell, so there is an isomorphism between them ibered over. Interchanging the pair o comma objects proves similarly that w w = id. The homotopy 2-category The strict 2-categories K 2 o interest arise as the homotopy 2-category o a qasicategorical context K. qasi-categorical context, is a simplicially enriched category whose hom-spaces are qasi-categories. qasi-category is a simplicial set with a weak composition o morphisms in all dimensions. Part o the point o this talk is that the precise deinition o this notion does not matter. qasi-categorical context comes eqipped with two distingished classes o maps, which we call eqivalences and isoibrations. The totality o this data satisies the properties enjoyed by the ibrant objects in a model category that is enriched over simplicial sets (with the Joyal model strctre) and in which all ibrant objects are coibrant. The prototypical example is given by the category o qasi-categories: the Joyal model strctre is monoidal. Other examples inclde complete Segal spaces or more general categories o Rezk objects: simplicial objects in a model category that are Reedy ibrant and satisy the Segal and completeness conditions. For instance, arwick s n-old complete Segal space model o (, n)-categories has this orm. I K is a qasi-categorical context, so is the slice category K/ over any object. Remark. Withot too mch additional diiclty, the hypothesis that all o the ibrant objects are coibrant cold be removed, bt we don t know o any interesting examples in which this is not the case. Deinition. The homotopy 2-category K 2 o a qasi-categorical context K is the strict 2-category deined by applying the homotopy category nctor ho: qcat Cat to each o the hom-spaces in K. Its objects are the objects o K; its 1-cells are the morphisms o K, which we call nctors; and its 2-cells are homotopy classes o 1-simplices in the hom-spaces o K. qasi-categorical context admits exponentials by arbitrary simplicial sets, and this strctre descends to 2-nctors on the homotopy 2-category o the orm described above. Moreover, the homotopy 2-category has weak comma objects g deined by the pllback g 1 C g in K. This pllback exists becase the ibrant objects in a simplicial model category are closed nder osield-kan-style homotopy limits. More precisely, the comma constrction is an example o a simplicially enriched (weighted) limit o a pointwise ibrant diagram whose weight is projective coibrant. Formal category theory in a qasi-categorical context. The content o the papers [RV1, RV2, RV3] is stated in the langage o qasi-categories bt all o the reslts appearing there apply, essentially withot change, in any qasi-categorical context K. What this means is that the deinitions o the basic categorical concepts

6 6 EMILY RIEHL (JOINT WORK WITH DOMINIC VERITY) can be interpreted in K and the proos, largely taking place in the homotopy 2- category, are also nchanged. For example, in [RV2], we deine the object o algebras or a homotopy coherent monad to be a particlar projective coibrant weighted limit and prove the monadicity theorem, which characterizes the accompanying ree-orgetl adjnction in K 2. Two-sided discrete ibrations Perhaps the most important thing that is missing rom the basic ramework o a 2-category with weak comma objects is the Yoneda embedding (classically, the hom binctor op Set) and its generalizations (arbitrary nctors op Set). These go by a variety o names: modles, pronctors, distribtors, or correspondences. There are several possible ways to ene modles in a 2- category. Given the strctres that are present in a homotopy 2-category K 2, or preerence will be to se weak comma objects. For example, the Yoneda embedding or is ened by the comma object: y 1-cell indction, any 2-cell with omain can be ened by a nctor abtting to. = t has additional niversal properties relating to the pre- and postcomposition actions by arrows in, which can only partially be established by 2-cell indction. These additional niversal properties are expressed by saying that is a two-sided discrete ibration in K 2. Cartesian ibrations. To state this deinition, we irst need a notion o cartesian ibration. For qasi-categories, this coincides exactly with the notion introdced by Lrie, bt or 2-categorical deinition can be interpreted in any homotopy 2- category. Deinition. n isoibration p: E is a cartesian ibration i (i) Every e E admits a p-cartesian lit χ: ē e along p. Here a 2- b α p cell χ is p-cartesian i it satisies a weak orm o the expected actorization axiom and also has a 2-cell conservativity property: any enorphism o χ siting over the identity on b is an isomorphism. (ii) The p-cartesian 2-cells are stable nder restriction along any nctor.

7 TOWRD THE FORML THEORY OF (, n)-ctegories 7 Replacing the 2-category K 2 by its dal K co 2, which reverses the 2-cells bt not the 1-cells, we obtain the notion o a cocartesian ibration. The search or examples is greatly aided by the ollowing theorem, which is a generalization o a 2-categorical reslt o Street [S]. Theorem. For an isoibration p: E, the ollowing are eqivalent: (i) p is a cartesian ibration. (ii) The natral nctor E p p 8 88 admits a right adjoint over. (iii) The natral nctor E E p admits a right adjoint right inverse. The right adjoint in (ii) picks ot the ain o the lited 2-cells; the p-cartesian lit is speciied by the conit. The nctor in (iii) picks ot the p-cartesian lit. Example. Using this theorem and the niversal properties o weak comma objects, it is relatively straightorward to show that : E E E is a cartesian ibration. More generally, : g is a cartesian ibration. Deinition. cartesian ibration p: E is discrete i any 2-cell E whose composite with p is an identity is an isomorphism. Example. Let 1 K 2 denote the terminal object. I b: 1 is a point, then : b is a discrete cartesian ibration. Finally: Deinition. span q E p is a two-sided discrete ibration i (i) E is a discrete cartesian ibration in K 2 /. (ii) E is a discrete cocartesian ibration in K 2 /. Example. The span C g is a two-sided discrete ibration. The eqipment or qasi-categories With the notion o a two-sided discrete ibration to ene modles, we can now introdce the complete ramework or ormal category theory. Theorem. There is a bicategory qmod 2 o qasi-categories; modles, i.e., twosided discrete ibrations q E p, written E : ; and isomorphism classes o maps o spans. Moreover, qmod 2 is biclosed: the nctors E and E admit right biadjoints. Note that the proposition proven above asserts that i, then the modles and are isomorphic as 1-cells. Remark. For this reslt, and rom hereon, we have specialized to the case o qasicategories. Twice in its proo, we se the act that (co)cartesian ibrations are homotopy exponentiable : that pllback along a (co)cartesian ibration p: E deines a let Qillen nctor p : sset/ sset/e. s in category theory, not all nctors have this property, bt there is a particlar condché condition, asserting that a certain amily o simplicial sets are weakly contractible, that implies it. We

8 8 EMILY RIEHL (JOINT WORK WITH DOMINIC VERITY) have yet to explore how it may be generalized to other contexts. For the time being, we might note that the strctres on qmod 2 reqiring this condition are convenient, bt not strictly necessary. The bicategory qmod 2 extends the homotopy 2-category o qasi-categories: Theorem. The identity-on-objects homomorphism qcat 2 qmod 2 that carries : to : is locally lly aithl. The proo o this reslt ses the Yoneda lemma or maps between modles. Theorem. The covariant represented modle : is let adjoint to the contravariant represented modle : in qmod 2. The preceding three theorems combine to assert that qcat 2 qmod 2 deines an eqipment or qasi-categories, in the sense o Wood [W]. Formal category theory. With this strctre in place, we can now commence with the ormal category theory. For example, there is a standard deinition o a right (Kan) extension diagram in any 2-category. In qcat 2 this is too weak (ailing, in general, to be pointwise ), bt in qmod 2 it gives the correct notion. Deinition. Consider a pair o nctors : and g : C. ecase qmod 2 is closed, we can orm the ollowing right extension diagram C g 7 C E I the modle E is covariantly representable, i.e., i E = r or some r : C, then r is the right extension o along g. We se the ollowing theorem to identiy representable modles, which allows s to determine when right extensions exist. Theorem. modle C q E p is covariant representable i and only i the ollowing eqivalent conditions hold: (i) q has a right adjoint right inverse (ii) each iber o q has a terminal object The right inverse o (i), composed with p, deines the representing nctor C. Condition (ii) can be sed to establish the expected reslt: i has limits indexed by certain comma objects, then the right extension o along g exists. Epiloge: the condché condition Consider an isoibration p: E between qasi-categories. We want a condition that implies that p : sset/ sset/e is let Qillen with respect to the Joyal model strctres. Deinition. p is condché i or all n 1 d k e E p n b 0 < k < n,

9 TOWRD THE FORML THEORY OF (, n)-ctegories 9 the condché space C p (e, b, k) is weakly contractible. The condché space C p (e, b, k) is a simplicial set whose vertices are those n- simplices o E lying over b and with kth ace e. The higher simplices are deined similarly bt with respect to degenerated copies o b at the kth vertex. For any isoibration p, the proo that p is let Qillen immediately redces to the qestion o whether the monomorphism Y deined by pllback Y E Λ n k p is a trivial coibration in the Joyal model strctre. Note that the vertices e indexing the condché spaces correspond to (n 1)-simplices that are present in bt missing rom Y. The 0-simplices o C p (e, b, k) are the n-simplices that cold be attached to a Λ n k-horn in Y to adjoin e. The proo that plling back along a condché nctor is let Qillen ses a Reedy category argment to present Y as a relative cell complex bilt rom inner horn inclsions indexed by data derived rom the condché spaces. Reerences [RV1] Riehl, E. and Verity, D. The 2-category theory o qasi-categories, (2013), 1 87, ariv: [RV2] Riehl, E. and Verity, D. Homotopy coherent adjnctions and the ormal theory o monads, (2013), 1 86, ariv: [RV3] Riehl, E. and Verity, D. Completeness reslts or qasi-categories o algebras, homotopy limits, and related general constrctions, (2014), 1 29, to appear in Homol. Homotopy ppl., ariv: [S] Street, R. Fibrations and Yoneda s lemma in a 2-category, Lectre Notes in Math. 420, (1974), [W] Wood, R.J. bstract proarrows I, Cahiers de Topologie et Géométrie Diérentielle Catégoriqes 23(3), (1982), n b